Question

Graph the function f(x) = x + 4 / x

Graph the secant line that passes through the points (1,5) and (8,8.5) on the same set of axes

Find the number c that satisfies the conclusion of the Mean Value Theorem for f on [1,8]

Answer 1

for the mean value theorm says that assuming that f(x) is continious over the range [1 8], and is differential over the same interval the the derivitave at c will have the same slope as the line made by the 2 given lines.

f'(x) = (f(b)-f(c))/(b-a)

Answer 2

so first take the deritive of f(x)

Answer 3

did you solve it yet ?

Answer 4

the derivative gives us an equation, a function, that we can use to define the slope at any point along the curve.

so, we need to know what the slope is between the points given to compare with

Answer 5

Wait, so do I have the sketch the whole graph ( therefore find the asymptote, critical point,etc...?)

Answer 6

y=f(x)

y' = f'(x) defines the slope at every value of x

given 2 points; (a,f(a)) and (b,f(b)) we determine the slope between them as\[\frac{f(b)-f(a)}{b-a}\]

there exists some point c in the interval from a to b such that the derivate defined at c is equal to to the slope of the secant between a and b\[f'(c)=\frac{f(b)-f(a)}{b-a}\]

Answer 7

what is the slope between (1,5) and (8,8.5) ??

Answer 8

So I would do this (2c)(c) - (c ^2) / c^2 = f(1) - (f8)/1-8 . Sorry my computer was derping I couldnt write the answer properly

Answer 9

yes,

but let me chk your derivative

(x + 4) / x

1+4x^(-1)

-4x^(-2) is what i get

Answer 10

nono it goes like this x + (4/x) so i simplified to x^2 +4/x so I used the product rule to give me the final answer?

Answer 11

(x + 4) / x; x=8 ... 12/8 not equal 8.5 murmur so its 2 terms

x + 4x^(-1)

1- 4x^(-2)

Answer 12

ok, so what the slope between the points given?

Answer 13

Hum you take the initial formula and I plugged in 1 and 8 to give me f(1) = 5 and f(8)= 3/2. THen I would need to use a quadratic formula to solve for c?

Answer 14

(8,8.5)
-(1,5.0)
--------
7,3.5 ; slope = 35/70 = 1/2

Answer 15

Shouldn't i use the point 1, and 8 and find the x for those points? or I have to use 8.5?

Answer 16

1 and 8 are the xs for those points; and 5 and 8.5 are the ys that result .. they already did that part for you

Answer 17

1 - 4c^(-2) = 1/2

- 4c^(-2) = -1/2

4/c^2 = 1/2

8 = c^2

Answer 18

Ah thank you it makes much more sense! thanks a lot!

Answer 19

youre welcome, good luck

Answer 20

notice that 2sqrt(2) is in the interval from 1 to 8, thereby validating the mean thrm